While moving head bone,mouth isnt moving along. I dont know what to do next
This is because the upper lip of the rat has more weight painting than the lower lip.  Thus, the bone has more influence on the upper lip than the lower.  If you want the whole mouth to move the same amount with the head, I would suggest you weight paint the entire mouth a value  of one.  You will have to do some smoothing from below the mouth down, however, so you still get a smooth deform.

The other problem you are having is that bones .004 and .015 have influence on the mouth.  If you remove their influence, that should make the entire mouth move with the head bone.